# convolution properties

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• 8/3/2019 Convolution Properties

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Convolution PropertiesDSP for Scientists

Department of PhysicsUniversity of Houston

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Properties of Delta Function

[n]: Identity for Convolution

x[n] [n] =x[n]

x[n] k[n] = kx[n]

x[n] [n + s] =x[n + s]

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Mathematical Properties of

Convolution (Linear System)

Commutative: a[n] b[n] = b[n] a[n]

a[n] b[n] y[n]

Then b[n] a[n] y[n]

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Properties of Convolution

Associative:{a[n] b[n]} c[n] = a[n] {b[n] c[n]}

If

a[n] b[n] c[n] y[n]

Then a[n] b[n] c[n] y[n]

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Properties of Convolution

Distributive

a[n]b[n] + a[n]c[n] = a[n]{b[n] + c[n]}

If b[n]

a[n] + y[n]

c[n]

Then

a[n] b[n]+c[n] y[n]

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Properties of Convolution

Transference: between Input & Output

Suppose x[n] * h[n] =y[n] IfL is a linear system,

x1[n] =L{x[n]}, y1[n] =L{y[n]} Then

x1[n] h[n]= y1[n]

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Continue

Ifx[n] h[n] y[n]

Linear Same Linear

System System

Then

x1[n] h[n] y1[n]

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Special Convolution Cases

Auto-Regression (AR) Model

y[n] = k = 0, M - 1.h[k]x[n - k]

For Example: y[n] =x[n] -x[n - 1] (first difference)

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Special Convolution Cases

Moving Average (MA) Model

y[n] = bx[n] + k= 1,M- 1b[k]y[n - k]

For Example: y[n] =x[n] +y[n - 1] (Running Sum)

AR and MA are Inverse to Each Other

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Example

For One-order Difference Equation (MAModel)

y[n] = ay[n - 1] +x[n]

Find the Impulse Response, if the system is

(a) Causal

(b) Anti-causal

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Causal System Solution

Input: [n] Output: h[n] For Causal system, h[n] = 0, n < 0

h = ah[-1] + = 1

h = ah +  = a

h[n] = anu[n]

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Anti-causal

Input:x[n] = [n] Output:y[n] = h[n] For Anti-Causal system, h[n] = 0, n > 0

y[n - 1] = (y[n] -x[n]) /a h = (h - ) /a = 0

h[-1] = (h -)/ a = - a

-1

h[- n] = - a-n h[n] = -anu[-n - 1]

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Central Limit Theorem

If a pulse-like signal is convoluted with

itself many times, a Gaussian will be

produced.

a[n] 0

a[n] a[n] a[n] a[n] = ???

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Central Limit Theorem

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Correlation !!!

Cross-Correlation a[n] b[-n] = c[n]

Auto-Correlation:

a[n] a[-n] = c[n] Optimal Signal Detector (Not Restoration)

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Correlation Detector

- 1 5 - 1 0 - 5 0 5 1 0 1 5

- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

- 2 5 - 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0 2 5- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

2

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Correlation Results

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

-6

-4

-2

0

2

4

6

8

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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Low-Pass Filter

Filter h[n]:

Cutoff the high-frequency components(undulation, pitches), smooth the signal

h[n] 0, (- 1)n h[n] = 0, n = -,

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Example: Lowpass

0 50 100 150 200 250 300 350-60

-40

-20

0

20

40

60

80

10 0

12 0

14 0

0 50 100 150 200 250 300 350

-60

-40

-20

0

20

40

60

80

10 0

12 0

14 0

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.1

0

0.1

0.2

0.3

0.4

0.5

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High-Pass Filter

Filter g[n]:

Remove the Average Value of Signal

(Direct Current Components), Only

Preserve the Quick Undulation Terms

ng[n] = 0

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Example: Highpass

0 50 100 150 200 250 300 350-8

-6

-4

-2

0

2

4

6

8

0 50 100 150 200 250 300 350

-60

-40

-20

0

20

40

60

80

10 0

12 0

14 0

-4 -3 -2 -1 0 1 2 3 4

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

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Delta Function

x[n] [n] =x[n]

Do not Change Original Signal Delta function: All-Pass filter

Further Change: Definition (Low-pass,

High-pass, All-pass, Band-pass )